Various approaches have been developed to recognize patterns associated with handwritten characters. Most handwriting recognition systems use recognizers based on Neural Nets, Hidden Markov Models (HMM) or a K-Nearest-Neighbor (KNN) approach. These systems perform reasonably well at the task of classifying characters based on their total appearance. For example, a level of similarity can be determined by generating a distance measure between patterns.
The advent of artificial intelligence within computer science has brought an abundance of decision-support systems. Decision-support systems are computer systems in which decisions, typically rendered by humans, are recommended and sometimes made. In creating decision-support systems, computer scientists seek to provide decisions with the greatest possible accuracy. Thus, computer scientists strive to create decision-support systems that are equivalent to or more accurate than a human expert. Applications of decision-support systems include medical diagnosis, troubleshooting computer networks, or other systems wherein a decision is based upon identifiable criteria.
One of the most promising new areas for research in decision-support systems is Bayesian networks. A Bayesian network is a representation of the probabilistic relationships among distinctions about the world. Each distinction, sometimes called a variable, can take on one of a mutually exclusive and exhaustive set of possible states. A Bayesian network is expressed as an acyclic-directed graph where the variables correspond to nodes and the relationships between the nodes correspond to arcs.
FIG. 1 depicts an example of a Bayesian network 101. In FIG. 1 there are three variables, X1, X2, and X3, which are represented by nodes 102, 106 and 110, respectively. This Bayesian network contains two arcs 104 and 108. Associated with each variable in a Bayesian network is a set of probability distributions. Using conditional probability notation, the set of probability distributions for a variable can be denoted by p(xi|Πi, ξ), where “p” refers to the probability distribution, where “Πi” denotes the parents of variable Xi and where “ξ” denotes the knowledge of the expert. The Greek letter “ξ” indicates that the Bayesian network reflects the knowledge of an expert in a given field. Thus, this expression reads as follows: the probability distribution for variable Xi given the parents of Xi and the knowledge of the expert. For example, X1 is the parent of X2. The probability distributions specify the strength of the relationships between variables. For instance, if X1 has two states (true and false), then associated with X1 is a single probability distribution p(x1|ξ) and associated with X2 are two probability distributions p(x2|x1=t, ξ) and p(x2|x1=t, ξ). In the remainder of this specification, ξ is not specifically mentioned.
The arcs in a Bayesian network convey dependence between nodes. When there is an arc between two nodes, the probability distribution of the first node depends upon the value of the second node when the direction of the arc points from the second node to the first node. For example, node 106 depends upon node 102. Therefore, nodes 102 and 106 are said to be conditionally dependent. Missing arcs in a Bayesian network convey conditional independencies. For example, node 102 and node 110 are conditionally independent given node 106. However, two variables indirectly connected through intermediate variables are conditionally dependent given lack of knowledge of the values (“states”) of the intermediate variables. Therefore, if the value for node 106 is known, node 102 and node 110 are conditionally dependent.
In other words, sets of variables X and Y are said to be conditionally independent, given a set of variables Z, if the probability distribution for X given Z does not depend on Y. If Z is empty, however, X and Y are said to be “independent” as opposed to conditionally independent. If X and Y are not conditionally independent, given Z, then X and Y are said to be conditionally dependent given Z.
The variables used for each node may be of different types. Specifically, variables may be of two types: discrete or continuous. A discrete variable is a variable that has a finite or countable number of states, whereas a continuous variable is a variable that has an uncountably infinite number of states. All discrete variables considered in this specification have a finite number of states. An example of a discrete variable is a Boolean variable. Such a variable can assume only one of two states: “true” or “false.” An example of a continuous variable is a variable that may assume any real value between −1 and 1. Discrete variables have an associated probability distribution. Continuous variables, however, have an associated probability density function (“density”). Where an event is a set of possible outcomes, the density p(x) for a variable “x” and events “a” and “b” is defined as:
      p    ⁡          (      x      )        =            Lim              a        →        b              ⁡          [                        p          ⁡                      (                          a              ≤              x              ≤              b                        )                                                          (                          a              -              b                        )                                        ]      where p(a≦x≦b) is the probability that x lies between a and b.
FIG. 2 depicts an example Bayesian network for troubleshooting automobile problems. The Bayesian network of FIG. 2 contains many variables 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, and 234, relating to whether an automobile will work properly, and arcs 236, 238, 240, 242, 244, 246, 248, 250, 252, 254, 256, 258, 260, 262, 264, 268. A few examples of the relationships between the variables follow. For the radio 214 to work properly, there must be battery power 212 (arc 246). Battery power 212, in turn, depends upon the battery working properly 208 and a charge 210 (arcs 242 and 244). The battery working properly 208 depends upon the battery age 202 (arc 236). The charge 210 of the battery depends upon the alternator 204 working properly (arc 238) and the fan belt 206 being intact (arc 240). The battery age variable 202, whose values lie from zero to infinity, is an example of a continuous variable that can contain an infinite number of values. However, the battery variable 208 reflecting the correct operations of the battery is a discrete variable being either true or false.
The automobile troubleshooting Bayesian network also provides a number of examples of conditional independence and conditional dependence. The nodes operation of the lights 216 and battery power 212 are dependent, and the nodes operation of the lights 216 and operation of the radio 214 are conditionally independent given battery power 212. However, the operation of the radio 214 and the operation of the lights 216 are conditionally dependent. The concept of conditional dependence and conditional independence can be expressed using conditional probability notation. For example, the operation of the lights 216 is conditionally dependent on battery power 212 and conditionally independent of the radio 214 given the battery power 212. Therefore, the probability of the lights working properly 216 given both the battery power 212 and the radio 214 is equivalent to the probability of the lights working properly given the battery power alone, P(Lights|Battery Power, Radio)=P(Lights|Battery Power). An example of a conditional dependence relationship is the probability of the lights working properly 216 given the battery power 212 which is not equivalent to the probability of the lights working properly given no information. That is, p(Lights|Battery Power)≠p(Lights).
There are two conventional approaches for constructing Bayesian networks. Using the first approach (“the knowledge-based approach”), a person known as a knowledge engineer interviews an expert in a given field to obtain the knowledge of the expert about the field of expertise of the expert. The knowledge engineer and expert first determine the distinctions of the world that are important for decision making in the field of the expert. These distinctions correspond to the variables of the domain of the Bayesian network. The “domain” of a Bayesian network is the set of all variables in the Bayesian network. The knowledge engineer and the expert next determine the dependencies among the variables (the arcs) and the probability distributions that quantify the strengths of the dependencies.
In the second approach (“called the data-based approach”), the knowledge engineer and the expert first determine the variables of the domain. Next, data is accumulated for those variables, and an algorithm is applied that creates a Bayesian network from this data. The accumulated data comes from real world instances of the domain. That is, real world instances of decision making in a given field. Conventionally, this second approach exists for domains containing only discrete variables.
After the Bayesian network has been created, the Bayesian network becomes the engine for a decision-support system. The Bayesian network is converted into a computer-readable form, such as a file and input into a computer system. Then, the computer system uses the Bayesian network to determine the probabilities of variable states given observations, determine the benefits of performing tests, and ultimately recommend or render a decision. Consider an example where a decision-support system uses the Bayesian network of FIG. 2 to troubleshoot automobile problems. If the engine for an automobile did not start, the decision-based system could request an observation of whether there was gas 224, whether the fuel pump 226 was in working order by possibly performing a test, whether the fuel line 228 was obstructed, whether the distributor 230 was working, and whether the spark plugs 232 were working. While the observations and tests are being performed, the Bayesian network assists in determining which variable should be observed next.
By utilizing both expert knowledge and empirical data, the network generator provides an improved Bayesian network that is more accurate than conventional Bayesian networks. In addition, the exemplary embodiment facilitates the use of continuous variables in Bayesian networks and handles missing data in the empirical data that is used to construct Bayesian networks.
Expert knowledge consists of two components: an equivalent sample size or sizes (“sample size”), and the prior probabilities of all possible Bayesian-network structures (“priors on structures”). The effective sample size is the effective number of times that the expert has rendered a specific decision. For example, a doctor with 20 years of experience diagnosing a specific illness may have an effective sample size in the hundreds. The priors on structures refers to the confidence of the expert that there is a relationship between variables (e.g., the expert is 70 percent sure that two variables are related). The priors on structures can be decomposed for each variable-parent pair known as the “prior probability” of the variable-parent pair. Empirical data is typically stored in a database. An example of acquiring empirical data can be given relative to the Bayesian network of FIG. 2. If, at a service station, a log is maintained for all automobiles brought in for repair, the log constitutes empirical data. The log entry for each automobile may contain a list of the observed state of some or all of the variables in the Bayesian network. Each log entry constitutes a case. When one or more variables are unobserved in a case, the case containing the unobserved variable is said to have “missing data.” Thus, missing data refers to when there are cases in the empirical data database that contain no observed value for one or more of the variables in the domain. An assignment of one state to each variable in a set of variables is called an “instance” of that set of variables. Thus, a “case” is an instance of the domain. The “database” is the collection of all cases.
An example of a case can more clearly be described relative to the Bayesian network of FIG. 2. A case may consist of the battery age 202 being 2.132 years old, the battery working properly 208 being true, the alternator working properly 204 being true, the fan belt being intact 206 being true, the charge 210 being sufficient, the battery power 212 being sufficient, the starter working properly 220 being true, the engine turning over 218 being true, the amount of gas 224 being equal to 5.3 gallons, the fuel pump working properly 226 being true, the fuel line working properly 228 being true, the distributor working properly 230 being false, the spark plugs working properly 232 being true and the engine starting 234 being false. In addition, the variables for the gas gauge 222, the radio working properly 214 and the lights working properly 216 may be unobserved. Thus, the above-described case contains missing data.